Nonharmonic Fourier series

Summary

A nonharmonic Fourier series is a series $\sum_n c_n e^{i\lambda_n t}$ on an interval, where the frequencies $\lambda_n$ need not be the integers. Ordinary Fourier series on $[-\pi, \pi]$ are the case $\lambda_n = n$. The main questions ask when the exponentials $e^{i\lambda_n t}$ are complete in $L^2$ of the interval. They also ask when they form a Riesz basis or a frame. The answers involve the density of the frequencies and entire functions of exponential type. They matter in sampling theory and control theory.